An entropy based measure for comparing distributions of complexity

• An entropy based measure for comparison of diversity of complexity is proposed.
• The measure allows for comparison of diversity both within and across distributions.
• The measure is multiplicative i.e., a doubling of value implies a doubling of diversity.

This paper is part of a series addressing the empirical/statistical distribution of the diversity of complexity within and amongst complex systems. Here, we consider the problem of measuring the diversity of complexity in a system, given its ordered range of complexity types ii and their probability of occurrence pipi, with the understanding that larger values of ii mean a higher degree of complexity. To address this problem, we introduce a new complexity measure called case-based entropy  CcCc — a modification of the Shannon–Wiener entropy measure HH. The utility of this measure is that, unlike current complexity measures–which focus on the macroscopic complexity of a single system–CcCc can be used to empirically identify and measure the distribution of the diversity of complexity within and across multiple natural and human-made systems, as well as the diversity contribution of complexity of any part of a system, relative to the total range of ordered complexity types.